An elementary proof for the exactness of (D, G) scaling

The goal of this paper is to provide an elementary proof for the exactness of the (D, G) scaling applied to the uncertainty structure with one repeated real scalar block and one full complex matrix block. The (D, G) scaling has vast application area around control theory, optimization and signal processing. This is because, by applying the (D, G) scaling, we can convert inequality conditions depending on an uncertain parameter to linear matrix inequalities (LMIs) in an exact fashion. However, its exactness proof is tough, and this stems from the fact that the proof requires an involved matrix formula in addition to the standard Lagrange duality theory. To streamline the proof, in the present paper, we clarify that the involved matrix formula is closely related to a norm preserving dilation under structural constraints. By providing an elementary proof for the norm preserving dilation, it follows that basic results such as Schur complement and congruence transformation in conjunction with the Lagrange duality theory are enough to complete a self-contained exactness proof.

[1]  A. Tits,et al.  Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics , 1991 .

[2]  A. Tits,et al.  Robustness in the Presence of Joint parametric Uncertainty and Unmodeled Dynamics , 1988, 1988 American Control Conference.

[3]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[4]  Dimitri Peaucelle,et al.  LMI-based Periodically Time-Varying Dynamical Controller Synthesis for Discrete-Time Uncertain Linear Systems , 2008 .

[5]  David J. N. Limebeer,et al.  Linear Robust Control , 1994 .

[6]  M. Fu,et al.  A dual formulation of mixed μ and on the losslessness of (D, G) scaling , 1997, IEEE Trans. Autom. Control..

[7]  A. Rantzer.  On the Kalman-Yakubovich-Popov lemma , 1996 .

[8]  T. Iwasaki,et al.  Generalized S-procedure and finite frequency KYP lemma , 2000 .

[9]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[10]  Venkataramanan Balakrishnan,et al.  Semidefinite programming duality and linear time-invariant systems , 2003, IEEE Trans. Autom. Control..

[11]  W. Kahan,et al.  NORM-PRESERVING DILATIONS AND THEIR APPLICATIONS TO OPTIMAL ERROR BOUNDS* , 1982 .

[12]  Pierre-Alexandre Bliman,et al.  A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters , 2003, SIAM J. Control. Optim..

[13]  M. Fu,et al.  A dual formulation of mixed μ and on the losslessness of (D, G) scaling , 1997, IEEE Trans. Autom. Control..

[14]  Carsten W. Scherer,et al.  LMI Relaxations in Robust Control , 2006, Eur. J. Control.

[15]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[16]  Shinji Hara,et al.  Generalized KYP lemma: unified frequency domain inequalities with design applications , 2005, IEEE Transactions on Automatic Control.